Empirically, we analyze weekly returns of the FTSE 100, NIKKEI 225 and Standard \(\&\)  Poor's 500 (S&P 500) market indices from April 1985 to February 2021.
In a first out-of-sample exercise we jointly estimate the VaR and ES of the three stock market indices using the proposed dynamic joint quantile regression model, hence taking into account for the correlation among the three indices. To evaluate VaR and ES forecasts and to show the main advantages of the proposed method, we use different backtesting procedures, where we compare the out-of-sample VaR and ES predictions with the ones obtained by applying the univariate method of [1]}. In particular, to perform a joint evaluation of VaR and ES, we follow [2]}, [3]}, [4]} and [1]} and extend their approach by introducing a new scoring function based on the MAL distribution. blackblackWe find that our multivariate method always provides more accurate VaR and ES predictions compared to other well-known approaches, like the Quantile AutoRegression of [6]} and the dynamic quantile regression of [1]}. Moreover, in line with [1]}, our results show that the Asymmetric Slope CAViaR specification of [9]} yields the best VaR and ES forecasts for all the three indices at different quantile levels, confirming the existence of relevant asymmetries in the impact of positive and negative returns.
